Problem 14
Question
Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$\left\\{\begin{aligned} x-4 y+z &=3 & & \\ y-3 z &=10 & & \text { Eliminate the } y \text { -term } \\ 3 y-8 z &=24 & & \text { from the third equation. } \end{aligned}\right.$$
Step-by-Step Solution
Verified Answer
The new system is: \( \begin{cases} x - 4y + z = 3 \\ y - 3z = 10 \\ z = -6 \end{cases} \).
1Step 1: Express y in terms of z using the second equation
The second equation is given as \( y - 3z = 10 \). To express \( y \) in terms of \( z \), solve for \( y \): 1. Add \( 3z \) to both sides: \[ y = 3z + 10 \]
2Step 2: Substitute y in the third equation
Substitute \( y = 3z + 10 \) into the third equation, \( 3y - 8z = 24 \):1. Replace \( y \) in \( 3y - 8z = 24 \) with \( 3z + 10 \):\[ 3(3z + 10) - 8z = 24 \] This will eliminate \( y \) from the equation.
3Step 3: Simplify the equation to remove y
Distribute and simplify the expression:1. Calculate \( 3(3z + 10) = 9z + 30 \)2. Substitute back into the equation: \( 9z + 30 - 8z = 24 \)3. Subtract \( 8z \) from \( 9z \): \( z + 30 = 24 \)4. Subtract 30 from both sides: \( z = -6 \)
4Step 4: New equivalent system without y-term in the third equation
Using the result from Step 3, write the new equivalent system by substituting \( z = -6 \) into the modified third equation.- First Equation remains unchanged: \[ x - 4y + z = 3 \]- Second Equation remains unchanged: \[ y - 3z = 10 \]- New form of the Third Equation: \[ z = -6 \]
Key Concepts
Elimination MethodSubstitution MethodEquivalent Systems
Elimination Method
The elimination method is a powerful tool for solving systems of linear equations. It involves creatively adding or subtracting equations to eliminate one variable, making it easier to solve the system.
- **Identify Target Variable:** Choose a variable to eliminate. In this case, it’s the \( y \)-term in the third equation.- **Match Coefficients:** Adjust the equations so that when one is subtracted from or added to another, a variable is canceled out. For instance, if the coefficients of \( y \) already match, you can directly add or subtract the equations.- **Simplify and Solve:** After eliminating one variable, simplify the resulting equation to make further calculations straightforward.
The elimination method makes solving systems of equations more efficient, especially when dealing with multiple linear equations. Practice makes perfect, so be sure to work on several problems to get comfortable with this method!
- **Identify Target Variable:** Choose a variable to eliminate. In this case, it’s the \( y \)-term in the third equation.- **Match Coefficients:** Adjust the equations so that when one is subtracted from or added to another, a variable is canceled out. For instance, if the coefficients of \( y \) already match, you can directly add or subtract the equations.- **Simplify and Solve:** After eliminating one variable, simplify the resulting equation to make further calculations straightforward.
The elimination method makes solving systems of equations more efficient, especially when dealing with multiple linear equations. Practice makes perfect, so be sure to work on several problems to get comfortable with this method!
Substitution Method
The substitution method excels in situations where one equation is already solved for a variable, or can be easily transformed to solve for one. It's about substituting one expression into another equation.
- **Solve for One Variable:** Start by solving one of the equations for a single variable. For example, from the second equation, express \( y \) as \( y = 3z + 10 \).- **Substitute into Another Equation:** Take the expression for \( y \) and substitute it into another equation. For instance, put \( y = 3z + 10 \) into the third equation.- **Simplify the Equation:** Substitute carefully and solve the resulting equation. Eliminate variables step-by-step until you find the solution for one variable.
The substitution method can be particularly useful when equations are complex, or coefficients might make elimination tricky. Understanding how to manipulate and substitute can simplify these problems significantly.
- **Solve for One Variable:** Start by solving one of the equations for a single variable. For example, from the second equation, express \( y \) as \( y = 3z + 10 \).- **Substitute into Another Equation:** Take the expression for \( y \) and substitute it into another equation. For instance, put \( y = 3z + 10 \) into the third equation.- **Simplify the Equation:** Substitute carefully and solve the resulting equation. Eliminate variables step-by-step until you find the solution for one variable.
The substitution method can be particularly useful when equations are complex, or coefficients might make elimination tricky. Understanding how to manipulate and substitute can simplify these problems significantly.
Equivalent Systems
Understanding equivalent systems is essential when working with linear equations. When performing operations like substitution or elimination, we transform our system into an equivalent system that retains the same solutions.
- **Same Solutions:** Two systems are equivalent if each equation in one can be derived from the equations of the other by algebraic manipulations. - **Operations:** Common operations that yield equivalent systems include adding, subtracting, or multiplying an equation by a number, and substituting variables. - **Preserve Relationships:** Ensure that any transformations maintain the relationships among equations.
In practice, checking if systems are equivalent involves simplifying equations and confirming they share solutions. Mastery of equivalent systems means you can confidently re-arrange equations, knowing you are solving the same underlying problem.
- **Same Solutions:** Two systems are equivalent if each equation in one can be derived from the equations of the other by algebraic manipulations. - **Operations:** Common operations that yield equivalent systems include adding, subtracting, or multiplying an equation by a number, and substituting variables. - **Preserve Relationships:** Ensure that any transformations maintain the relationships among equations.
In practice, checking if systems are equivalent involves simplifying equations and confirming they share solutions. Mastery of equivalent systems means you can confidently re-arrange equations, knowing you are solving the same underlying problem.
Other exercises in this chapter
Problem 14
Find the partial fraction decomposition of the rational function. $$\frac{x+6}{x(x+3)}$$
View solution Problem 14
Find the inverse of the matrix if it exists. $$\left[\begin{array}{lll}4 & 2 & 3 \\ 3 & 3 & 2 \\ 1 & 0 & 1\end{array}\right]$$
View solution Problem 14
Graph the inequality. $$x^{2}+(y-1)^{2} \leq 1$$
View solution Problem 14
Solve the matrix equation for the unknown matrix \(X\) or explain why no solution exists. $$\begin{aligned} &A=\left[\begin{array}{ll} 4 & 6 \\\1 & 3\end{array}
View solution